by noting \(\mathbf {u}^{(3,2)}_l = \mathbf {u}_{3,l} - \mathbf {u}^{(3,1)}_l\)  and \(\mathbf {u}^{(1,2)}_l = \mathbf {u}_{1,l} - \mathbf {u}^{(1,3)}_l\) . Thus, each unit \(l\)  spans a subspace of dim-4.
From Lemma REF  in Appendix , we see that the dimension of \({\rm span}\left({\bf H}_1{\bf U}_{H_1},{\bf H}_2{\bf U}_{H_2},{\bf H}_3{\bf U}_{H_3}\right)\)  is \(\min \left(2(3M-N),N\right)\) . Therefore, we can construct \(\min \left(\frac{3M-N}{2}, \frac{N}{4}\right)\)  linearly independent units following Pattern \(1.3\) .Here we assume that \(\frac{3M-N}{2}\)  is an integer. Otherwise, we use symbol extension [1]} to ensure that the dimension of the above intersection is dividable by two; see Appendix  for details. Note that the symbol extension is used to achieved a fractional DoF throughout of the rest of this paper without further explicit notification.
